( 47 ) 



- 2 (1 - 2yy + 2/' (1 - 3//)' + 3^^ (1 - 2y) {y^ + 2y - 1) = O, 

 or 3t/^ — \hy' -\- 2%' — 27?/' + 12?/ — 2 = O, 



i. e. after division by (?/ — 1)' : 



Zy- -6^ + 2, 

 yielding : 



2/ = 1 ± 7, 1/3". 



As it is obvious that y cannot be larger than 1, only : 

 y ^ 1 - Vs i/3 = 0,4226 

 satisfies here. 



If we substitute the value x -\- (p from {a!) into (c), we get: 



(1— 2v)' 



.t' (1 — x) — (1 — 2^0' -^ -^~ = 0. 



' ^ ' ^ /(l-3# 



In this the last fraction passes into V4 (1 + 1^3), after substitution 

 of ?/ = 1 — Vs V/3, so that we get for x : 



.^(l-.r)- 7.(1 + 1/3) jl_4.^.(l-.^.)j := 0, 

 hence : 



X{ 1 - .r) rz: 7, (_ 1 + 1/3), 



giving: 



,v = 7J 1 ± 7, (1/6 — \/2)\= 0,2412 or 0,7588. 



It is obvious from the figure, that only the first value satisfies, 

 viz.: 



^ = 7, i 1 - 7, {[/6 — [/2)\= 0,2412. 



The value of (f is finally found from (c) : 



x(l—x) 



(^ + <py = ' ' = 'A (2 + ^/3), 

 y (1—22/) 



giving x^(p = 7, (3 ^/2 + [/6), hence y=7,(— l + i/2 + |/6)=l,432. 



As y = 1 — 7, 1/3, to = 7, 1/3, i. e. the intersection takes place 

 ?it vz=b[/3 = 1,732 6. 



As mentioned before T„ = 2\ for y = 1,30 (see § 3). For (p = 1,43 

 T„ is already < ï\. For ^\/t, = ''/.^ (p' we find easily the value 

 1,215, while 2,887 is found for t,/t, = (1 + l/cpY. 



7. Besides the cases, given in figs. 1 and 2, representing the 

 principal types I and III, there is another important type, viz. II, 

 of which I also gave a full description in my previous paper, which 

 I have already cited several times 0. The ^,7'-diagram of this case 



1) 1. c. p. 663—667. 



